The various early texts regarding the battle at Bannockburn are varied as to the details of the battle. One problem is that none were contemporary accounts. They were all written well after the battle, in some cases decades later. The closest is that by Sir Thomas Gray, an English knight taken prisoner at Bannockburn who, with time on his hands, compiled the “Scalacronica.” An excerpt from that source is at: http://www.deremilitari.org/RESOURCES/SOURCES/scalacronica.htm  http://www.deremilitari.org/RESOURCES/SOURCES/scalacronica.htm Notice that here, “Peris de Mountforth, knight, was slain with an axe by the hand of Robert de Bruce, as was reported.” So according to the earliest chronicle, an actual participant of the battle, reports the knight in question was not a Henry de Bohun but rather a Piers de Montfort. Since he added ‘as reported’ I would take that to mean either he got the information from another prisoner, or that he himself saw it and it was verified by someone else.
Another English source, the Vita Edwardi Secundi also documents the incident: http://www.nls.uk/scotlandspages/timeline/1314.htmlhttp://www.nls.uk/scotlandspages/timeline/1314.htmlHere the anonymous author calls him: “a certain knight, Henry de Boun [Bohun].” Here, the author does not attempt to identify who his father or uncle were, as the Scottish sources did. It’s entirely possible the young knight was not related to the Earl of Hereford, Humphry de Bohun, that he simply had a similar or same surname, or...
Ronald McNair Scot, Robert the Bruce, 1992, pp. 152-3 also details the incident, but adds a valuable clue—that the English knight who challenged Bruce wore the crest of the de Bohuns on his surcoat. If this was in fact true, anyone seeing this might have assumed he was a de Bohun while he may have simply been a knight in the service of Humphry de Bohun.
Another problem, ES III, Tafel 688 lists another de Bohun family in England at that time—de Bohun of Midhurst and Eastbourne, descended (in the male line) from the Viscounts of Maine. One member, Svari fitzChaine of Midhurst, married Muriel, daughter of Richard, sn de Bohun, and took the name de Bohun. One later member of this family, Sir Humphrey de Bohun, was in 1454 Sheriff of Essex and Hertshire. While this table does not list a Henry of this particular family, it remains possible that the knight in question, if he were a de Bohun, may have been a member of a different de Bohun (or similar spelling) family. Look again at the Vita Edwardi Secundi where he is called Henry de Boun, and were “Bohun” was added in parentheses.
All in all it is far less than certain he was a de Bohun and of the de Bohun family of Hereford. I think on this issue ‘we just don’t know’ might have to suffice.